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(B) Let $f$ be an increasing function and $g$ be a decreasing function. Consider the composition $(fog)(x) = f(g(x))$. Let $x_1 < x_2$. Since $g$ is a

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Decreasing function

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(B) Let $f$ be an increasing function and $g$ be a decreasing function. Consider the composition $(fog)(x) = f(g(x))$. Let $x_1 Since $g$ is a decreasing function, $g(x_1) > g(x_2)$. Let $y_1 = g(x_1)$ and $y_2 = g(x_2)$. Then $y_1 > y_2$. Since $f$ is an increasing function, $f(y_1) \geq f(y_2)$ (or $f(y_1) > f(y_2)$ if $f$ is strictly increasing). Therefore, $f(g(x_1)) \geq f(g(x_2))$. Since $x_1 < x_2$ implies $(fog)(x_1) \geq (fog)(x_2)$, the function $fog$ is a decreasing function.

If $f$ is an increasing function and $g$ is a decreasing function, and $fog$ is defined, then what kind of function is $fog$?

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